Возможный подход с использованием Flexbox
body, html {
margin: 0;
padding: 0;
}
section {
height: 100vh;
display: flex;
flex-flow: column nowrap;
}
section > div {
padding: 20px;
}
div:not([id]) {
background: #d8d8dc;
flex: 1;
overflow: auto;
}
<section>
<div id="dont-scroll">
Unknown content! <br />
Lorem ipsum sit dolor amet
</div>
<div>
<div id="scroll-this">
This part should be scrollable when
the total content overflows the SECTION
<br /><br />
<p>Lorem ipsum dolor sit amet,
consectetuer adipiscing elit. Donec
odio. Quisque volutpat mattis eros.
Nullam malesuada erat ut turpis.
Suspendisse urna nibh, viverra non,
semper suscipit, posuere a, pede.</p>
...
</div>
</div>
</section>
Пример использования codepen